Properties

Label 304.2.h.b
Level $304$
Weight $2$
Character orbit 304.h
Analytic conductor $2.427$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{2} q^{7} + 4 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{2} q^{7} + 4 q^{9} + 2 \beta_{2} q^{11} + \beta_{3} q^{13} + 3 q^{17} + ( -\beta_{1} + 2 \beta_{2} ) q^{19} + \beta_{3} q^{21} + 3 \beta_{2} q^{23} -5 q^{25} + \beta_{1} q^{27} + \beta_{3} q^{29} -2 \beta_{1} q^{31} -2 \beta_{3} q^{33} -2 \beta_{3} q^{37} -7 \beta_{2} q^{39} + 2 \beta_{3} q^{41} -2 \beta_{2} q^{47} + 4 q^{49} + 3 \beta_{1} q^{51} -\beta_{3} q^{53} + ( -7 - 2 \beta_{3} ) q^{57} + 3 \beta_{1} q^{59} -8 q^{61} -4 \beta_{2} q^{63} + \beta_{1} q^{67} -3 \beta_{3} q^{69} -6 \beta_{1} q^{71} -11 q^{73} -5 \beta_{1} q^{75} + 6 q^{77} + 2 \beta_{1} q^{79} -5 q^{81} + 4 \beta_{2} q^{83} -7 \beta_{2} q^{87} + 4 \beta_{3} q^{89} -3 \beta_{1} q^{91} -14 q^{93} + 2 \beta_{3} q^{97} + 8 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 16q^{9} + O(q^{10}) \) \( 4q + 16q^{9} + 12q^{17} - 20q^{25} + 16q^{49} - 28q^{57} - 32q^{61} - 44q^{73} + 24q^{77} - 20q^{81} - 56q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/7\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} + 7 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 14 \nu \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{2} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\(7 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
303.1
1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i
−1.32288 + 2.29129i
0 −2.64575 0 0 0 1.73205i 0 4.00000 0
303.2 0 −2.64575 0 0 0 1.73205i 0 4.00000 0
303.3 0 2.64575 0 0 0 1.73205i 0 4.00000 0
303.4 0 2.64575 0 0 0 1.73205i 0 4.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.h.b 4
3.b odd 2 1 2736.2.k.k 4
4.b odd 2 1 inner 304.2.h.b 4
8.b even 2 1 1216.2.h.c 4
8.d odd 2 1 1216.2.h.c 4
12.b even 2 1 2736.2.k.k 4
19.b odd 2 1 inner 304.2.h.b 4
57.d even 2 1 2736.2.k.k 4
76.d even 2 1 inner 304.2.h.b 4
152.b even 2 1 1216.2.h.c 4
152.g odd 2 1 1216.2.h.c 4
228.b odd 2 1 2736.2.k.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.b 4 1.a even 1 1 trivial
304.2.h.b 4 4.b odd 2 1 inner
304.2.h.b 4 19.b odd 2 1 inner
304.2.h.b 4 76.d even 2 1 inner
1216.2.h.c 4 8.b even 2 1
1216.2.h.c 4 8.d odd 2 1
1216.2.h.c 4 152.b even 2 1
1216.2.h.c 4 152.g odd 2 1
2736.2.k.k 4 3.b odd 2 1
2736.2.k.k 4 12.b even 2 1
2736.2.k.k 4 57.d even 2 1
2736.2.k.k 4 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{2} - 7 \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -7 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 3 + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( ( 21 + T^{2} )^{2} \)
$17$ \( ( -3 + T )^{4} \)
$19$ \( 361 + 10 T^{2} + T^{4} \)
$23$ \( ( 27 + T^{2} )^{2} \)
$29$ \( ( 21 + T^{2} )^{2} \)
$31$ \( ( -28 + T^{2} )^{2} \)
$37$ \( ( 84 + T^{2} )^{2} \)
$41$ \( ( 84 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( ( 12 + T^{2} )^{2} \)
$53$ \( ( 21 + T^{2} )^{2} \)
$59$ \( ( -63 + T^{2} )^{2} \)
$61$ \( ( 8 + T )^{4} \)
$67$ \( ( -7 + T^{2} )^{2} \)
$71$ \( ( -252 + T^{2} )^{2} \)
$73$ \( ( 11 + T )^{4} \)
$79$ \( ( -28 + T^{2} )^{2} \)
$83$ \( ( 48 + T^{2} )^{2} \)
$89$ \( ( 336 + T^{2} )^{2} \)
$97$ \( ( 84 + T^{2} )^{2} \)
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